# Racing Brakes Numerical Analysis 1 – Introduction of the problem

**Abstract
**

** **This is the first writing in what will be a series of papers on the numerical analysis of brake rotors used in race cars. The purpose of this writing is to describe the goal of this series, to introduce the problem, its background, and the method that will be used to solve it. The most basic initial model will be introduced, along with a list of simplifying assumptions that were used, and the initial results it produced. Being the first writing in a series, the topic is covered more broadly, and many aspects of the topic are introduced at once. This resulted in a longer writing than what is anticipated for future writings.

A detail of the mathematical model will be released at a later date.

**Summary
**

** **The goal of this paper was to introduce the topic of the series and release preliminary results. The limits of the current code were mentioned, and the effects were discussed. Iterations, error, maximum temperature, and temperature distribution were observed with changes in relaxation, step size, and residual.

The maximum max temperature was 1,166K while the minimum max temperature was 1,158.7, a deviation of only .6%. It was observed that adding a relaxation factor significantly improved performance. For a residual of E-5 and a step size of .00025m the version without a relaxation factor took 188,642 iterations while with a relaxation factor only 9.905 iterations were necessary. The initial temperature guess also had a large effect. By guessing 1000K across the rotor before the first iteration, 22,294 iterations were necessary, while using the average temperature of each row as the guess for that row before the first iteration required only 16,264 iterations.

Error was somewhat erratic without relaxation, but with relaxation it was found that error was directly related to step size, and the size of the grid. It was found that the solution was first order accurate, and that as residual was decreased error exactly reached a value expected of a first order accurate system. The exact error was reached at a residual of E-6 for every step size.

The study was highly limited due to the initial assumption that the brake rotor was a slab. Though it provided a benefit in simplicity for troubleshooting the initial version of the code, it undoubtedly compromised the quantative results. Other assumptions were limiting factors, such as constant thermal conductivity with temperature.

**Introduction
**

Objectives

Besides introducing the topic, the goal is to obtain a base numerical analysis in order to have a means of later comparison. Values such as maximum and minimum temperatures are not to be treated as indicative of those seen in real life this early in the development of the engineering code. Rather, they will be used as a benchmark to observe how much a given change alters the results. Until the system has been converted to a radial, rather than the current model which must be understood by the reader to be an oversimplified model, any quantitative results of this initial analysis are only for later comparison.

The effects of changes in grid size and residual will be observed, as will the effects of a relaxation factor. Of particular interest will be the effects on the number of iterations, the time of the solution, the changes in maximum temperature, and the amount of error when compared to the conservation of energy. The temperature distribution of the model will also be analyzed.

Background

Brakes are used in automobiles to convert kinetic energy in to internal or thermal energy. In race cars, the purpose is to slow the car as quickly as possible to a speed that will enable the race vehicle to negotiate the looming corner. Due to the need to eliminate kinetic energy as quickly as possible, the levels of heat created can be unwieldy. It quickly becomes apparent that the braking system must be made to properly absorb all this heat, and improvements are sought constantly.

Figure 1 – LMP Brakes

There are several ways to improve a braking system’s ability to absorb heat, some of which will be mentioned before the method investigated by this series is described. The two most obvious methods to improve brake performance from a heat standpoint are to create materials more capable of coping with the high temperatures encountered in racing, and improving the cooling of the brakes. Some other methods involve segregating heat from the braking fluid, as well as raising the boiling point of the brake fluid.

This series will focus on the heat distribution through the brake rotor, and aim to reduce that temperature (particularly the maximum temperature) through geometry and airflow. Certainly, cooling can be increased through increased heat radiation, and other properties such as improved thermal conductivity, however these aspects are beyond the current scope of this series of writings.

Papers written for this series will follow a common format. Generally, when a new paper is released in the series its purpose will be to eliminate a previous simplifying assumption, fix an error, or improve the speed and accuracy of the code. There are many assumptions in this initial model which severely limit the results, and despite the slow elimination of these assumptions they will be prevalent in the writings. Mistakes will also be common. Some posts will contain no direct changes, but instead may detail the effects things like of grid size, and speed of convergence.

The purpose of this series is for the writer to start from a very basic and broad undergraduate engineering background, and slowly improve beyond the typical undergraduate simplified methods in a way that shows the reader the full process with as much transparency as possible. In no other conceivable way is this writing unique. The problems posed have already been solved by individuals with more experience in the field than I ever expect to achieve. The mathematical methods used already exist. In fact the only methods used that aren’t already well known will be fudge factoring and seat of the pants thinking. Even some of those methods though will likely already exist.

The reader is cautioned to keep all these points in consideration when deciding how to apply the content of this series.

**Model
**

Physical model

In order to simplify initial development and troubleshooting of the engineering code the physical model currently wasn’t a disk, but a giant block or slab. The cross sectional area of the block was the same as the cross section of the brake rotor. This cross section was then split in half at the midplane between the two planar faces of the rotor. This was done due to the assumed symmetry of the system, which is not the worst assumption and will remain with the code for the foreseeable future.

This boundary of symmetry was treated as adiabatic. Such a boundary condition is common practice, since it reduces the size of the model, makes for simpler equations, and does not increase error. The boundary conditions on the other three faces allow for convection and radiation. Radiation was assumed to contribute enough to the results to be included, and it was not difficult to add. Finally, there exists the region on the outer planar face where the brake pad creates friction with the rotor and generates heat. It was assumed that no heat transmits through the brake pad, and that the brake pad does not block escaping heat.

Figure 2 – Physical Model

The incoming heat was estimated from observations of the hairpin at Sebring International Raceway during the 12 Hours of Sebring 2012. It was estimated that LMP cars can decelerate at approximately 3 G’s, begin braking at 80m/s, and stop braking at 22.7m/s. The time taken, considering constant deceleration, was 1.95 seconds. It was assumed that 100% of the kinetic energy of the vehicle was converted to internal energy in the brake rotors. This amount of energy was then divided by ten, since it was estimated that the duty cycle of such braking was approximately 10%. Heat per rotor was then found by dividing the value by 4, which made the assumption that each brake rotor absorbed equal heat. In reality, the front rotors would encounter more heat than the rear rotors in general. It was decided that for this initial test, the incoming heat to the representative block would be 16,975 Watts. It was assumed that the heat generation would be constant across the contact area.

The brake rotor was discretized with equal step sizes in both directions (see Fig. 3) in the cross sectional area of the brake rotor, with a 1m depth for simplicity. The vertical steps symbolized infinitesimal steps in radius, and horizontal steps symbolized infinitesimal steps in the axial direction. This grid of nodes was represented in Matlab as a matrix, with i and j representing the location of the node, and the value in the ith column and jth row represented the temperature at that node.

Figure 3 – Discretization

The math model was based on the conservation of energy. Since the model was treated as steady state, the sum of energy entering and exiting any region of the brake rotor must equal zero.

**Results and discussion
**

** **The effectiveness of the relaxation factor was profound. The longest duration analysis that was run with and without a relaxation factor was a step size of .00025m (this is why most figures are for data collected with a step size of .00025), which resulted in a grid size of 40,000 nodes. For a residual of E-5 (which is the maximum residual for non relaxation factor runs) the solution took 450s and 188,642 iterations to converge (see Fig. 4). On the other hand, with an optimized relaxation factor the solution took 35s (almost 13 times less time) and 9,905 iterations (19 times fewer iterations). 419 iterations per second were achieved without the relaxation factor, whereas only 283 iterations a second were achieved with the relaxation factor. This makes sense, since including the relaxation factor increases the complexity of the equations, and therefore more calculations per iteration. However, due to the vast improvement in iteration count to convergence, using the relaxation factor clearly improves the speed of convergence.

Figure 4 – Iterations as a function of Residual, taken at .00025m step size

Figure 5 – Iterations as a function of step size, taken at E-5 Residual

Interestingly there was an improvement in error when using a relaxation factor for moderate residuals. However as the residual was reduced further, the error increased whereas under the same conditions the error decreased without the relaxation factor. It was found that the solution was first order accurate; meaning that halving the step size had the effect of halving the error. This was a desirable outcome since it validated the numerical method used was functional and none of the long equations used were accidently mistyped.

Figure 6 – Error as a function of Residual, taken at step size of .00025m

Figure 7 – Error as a function of step size, taken at maximum Residual run (E-5 to E-7)

The maximum temperature behaved in general as was expected. With large step sizes, the maximum temperature was high, and with smaller step sizes the maximum temperature was reduced. The absolute maximum temperature predicted was 1,166K and the minimum predicted by the code was 1,158.7. This made for a deviation of only .6% which is pretty good considering that there were two methods of solving (relaxation and without) as well as a step size range of .001 to .0001, and a residual range from E-3 to E-10. Figure 8 shows the maximum temperature as related by the step size, and each point was taken at maximum residual for that run, which was not constant. Figure 9 illustrates the change in maximum temperature prediction with respect to the residual.

Figure 8 – Maximum temperature as a function of step size, taken at minumum residual

Figure 9 – Maximum temperature as a function of residual, taken at .00025m step size.

The temperature distribution predicted by the code was as expected. The centerline shows the adiabatic boundary condition, where the rate of change in temperature with respect to movement in the horizontal was zero. The maximum temperature was found near the center of the pad. Outside of the direct influence of the heat flux from the brake pads, it can be seen that convection on the planar faces of the rotor is providing a cooling effect. This can be seen most notably in the lower corners of Figure 10.

Figure 10 – Temperature distribution of the slab cut in cross section

The initial temperature guess had a large effect on the number of iterations, as expected. Three methods were used to provide the initial temperature distribution guess. First (Fig. 11) used a simple 1000K guess for the whole rotor, which resulted in 22,294 iterations. Second, the average temperature from a previous run was applied across the entire rotor (Fig. 12), which took 21,092 iterations to converge. Last, the average temperature at each vertical location (representing changes in radius) was distributed across each row. This method resulted in only 16,264 iterations (Fig. 13). It was found that good prediction of temperature distribution can reduce iterations substantially. In this case, the iterations were reduced by 27%.

Figure 11 – Number of iterations resulting from 1000K initial guess

Figure 12 – Number of iterations resulting from T_{ave} initial guess

Figure 13 – Number of iterations from the vertical gradient of average temperatures for each row

**Conclusion and recommendations
**

** **It was concluded that the code itself is functional and capable of producing the results of the values of interest. Methods to reduce error and run time were observed, and will be implemented in to future code updates. Though the values predicted by the code are likely not indicative of real life values, they will still provide a benchmark for the near future when observing changes produced by differences in duty cycle, or by making material properties such as thermal conductivity vary with temperature as they do in real life.

Limits of this writing are numerous. First and foremost is the fact that the brake rotor for the time being is treated as a block or slab rather than a cylindrical body. Still another is the solution time for runs without a relaxation factor prohibited gathering data at step sizes smaller than .00025m as well as residuals beyond E-5. This made comparisons between relaxation and non relaxation less than ideal.

Future treatments of this subject should revise the geometry in order to properly represent the cylindrical nature of brake rotors. Time could be spent running the long simulations necessary to gather all data for non relaxation solutions, however since the model has so many other shortcomings this should be low on the priority list. While the model is this simple, making thermal conductivity and emissivity variable with temperature could be easy changes to implement in order to observe the effects on the slab.

**Data**

Figure 14 – Relaxation data

Figure 15 – No relaxation data

**Acknowledgements
**

** **Thanks for all the input from the forums at F1Technical.net.

**References
**

[1] Incropera et all, Fundamentals of Heat and Mass Transfer 6^{th} Ed., Wiley

[2] C. Gerald, P Wheatley, Applied Numerical Analysis 7^{th} Ed., Pearson Addison Wesley

[3] D. Pitts, L. Sissom, Heat Transfer 2^{nd} Ed., Schaum’s Outlines, McGraw Hill

[4] R.W. Hamming, Numerical Methods for Scientists and Engineers 2^{nd} Ed., Dover Publications

[5] F. Talati, S. Jalalifar, Investigation of heat transfer phenomena in a ventilated disk brake rotor with straight radial rounded vanes, Journal of Applied Sciences 8 (20) (2008) 3583-3592

[6] J. Zhang, Numerical Accuracy, tamu.edu, *https://ceprofs.civil.tamu.edu/jzhang/cven-302/chap04b.ppt*